Dynamical systems and forward-backward algorithms associated with the sum of a convex subdifferential and a monotone cocoercive operator

TL;DR

This paper introduces new continuous and discrete dynamical systems for solving structured monotone inclusions involving convex subdifferentials and cocoercive operators, analyzing their convergence and deriving novel forward-backward algorithms.

Contribution

It extends dynamical systems frameworks to include structured monotone operators and develops new forward-backward splitting methods with proven convergence.

Findings

Convergence of the proposed dynamical systems is established via Lyapunov analysis.

New forward-backward algorithms are derived from the discretization of continuous dynamics.

The methods are applicable to structured monotone inclusions with non-potential terms.

Abstract

In a Hilbert framework, we introduce continuous and discrete dynamical systems which aim at solving inclusions governed by structured monotone operators $A=\partial\Phi+B$, where $\partial\Phi$ is the subdifferential of a convex lower semicontinuous function $\Phi$, and $B$ is a monotone cocoercive operator. We first consider the extension to this setting of the regularized Newton dynamic with two potentials. Then, we revisit some related dynamical systems, namely the semigroup of contractions generated by $A$, and the continuous gradient projection dynamic. By a Lyapunov analysis, we show the convergence properties of the orbits of these systems. The time discretization of these dynamics gives various forward-backward splitting methods (some new) for solving structured monotone inclusions involving non-potential terms. The convergence of these algorithms is obtained under classical step size limitation. Perspectives are given in the field of numerical splitting methods for optimization, and multi-criteria decision processes.